Having the same problem. Posted this in another therad on the issue:
After moving the plug around, I can get it to charge/connect to my computer if I push it towards the phone and pull up, but only with one side of the connector. The other side never connects, no matter what angle of force is applied.
Physically, it must be an issue with the pins in the iPhone connecting fully with the pins on the Lightning connector. Because all Lightning connectors must be near-identical, I unfortunately think that it's a problem with the phone itself.
I'm bummed. I may fool around with it a bit more to see if I can gather more data, but it looks like I'll be scheduling for a replacement when Apple Stores near me get another shipment. Until then I'll have to fashion a rubberband apparatus for it to charge.
Please, if anyone reading this is also having this problem, post as much detail as you can about the problem. I would love this to just be an issue with the Lightning cable, but, we'll see.